Introduction to Quantum Chromodynamics

Introduction to Quantum ChromodynamicsLecture 3

Steffen Schumann

Institute for Theoretical Physics

University of Gottingen

Quy Nhon – MCnet school 201917/09/19

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http://www.physik.uni-goettingen.de

http://www.fulbright.de

http://www.montecarlonet.org

http://www.bmbf.de

http://physik.uni-goettingen.de



NLO QCD predictions

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Hard Processes at Next-to-Leading Order QCDThe need for higher precision

derive exclusion bounds from a given measurementnon-observation of a given hypothesis, e.g. heavy Higgs, BSM modelcrucial to know the corresponding production cross sections preciselyderive limits on particle masses, model parameters

counting experiments to extract certain parameters, couplings; at hadron colliders αS corrections indispensible, need one-loop QCD

stop mass bounds from Tevatron[from the early days, T. Plehn private communication]

10-1

1

100 110 120 130 140 150

∆MLO

∆MNLO

σNLO

σLO

: µ=[m(t)/2 , 2m(t)]

: µ=[m(t)/2 , 2m(t)]

m(t1) [GeV]

σ(pp– → t

1t−

1 + X) · BR(t

1t−

1 → ee + ≥ 2j) [pb]

CDF 95% C.L. upper limit (prel.)

m(χ1o)=m(t

1)/2

Higgs coupling extractionσ(gg → h→ γγ) ∼ Γg Γγ

Γ

σ(gg → h→WW ∗) ∼ Γg ΓWΓ

σ(qq → qqh, h→ ττ) ∼ ΓW ΓτΓ

...; Γi ∼ giih, Γg ∼ gtth; theoretical uncertainties dominate; systematics reduced for σ ratios

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Hard Processes at Next-to-Leading Order QCDAnatomy of NLO QCD calculations

|M|2 = |MB |2 + αS

(|MR |2 + 2<(MBM†V )

)�

γ∗/Z0

�γ∗/Z0

�γ∗/Z0

σNLO2→n =

∫n

d(4)σB +∫

n+1d(d)σR +

∫n

d(d)σV︸︷︷︸αS real & virtual corrections

real-emission σR

; IR divergence for soft/collinear emission(UV renormalized) virtual-corrections σV

; IR divergent when propagator goes on-shelldivergences manifest in dimensional regularisation, d = 4− 2ε; occurence of single & double poles, i.e. 1/ε, 1/ε2

for IR safe observables poles cancel, the sum is finite [Kinoshita ’62; Lee, Nauenberg ’64]

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|M|2 = |MB |2 + αS

(|MR |2 + 2<(MBM†V )

)�

γ∗/Z0

�γ∗/Z0

�γ∗/Z0

σNLO2→n =

∫n

d(4)σB +∫

n+1d(d)σR +

∫n


real-emission σR




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|M|2 = |MB |2 + αS

(|MR |2 + 2<(MBM†V )

)�

γ∗/Z0

�γ∗/Z0

�γ∗/Z0

σNLO2→n =

∫n

d(4)σB +∫

n+1d(d)σR +

∫n


real-emission σR




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Hard Processes at Next-to-Leading Order QCDInfrared & Collinear safe observables

interested in observable-independent formulation

OB =∫|M(n)

B |2 dΦnΘ(n)(O, {pn})

OR =∫|M(n+1)

R |2 dΦn+1Θ(n+1)(O, {pn+1})

OV =∫

2<(MBM†V) dΦnΘ(n)(O, {pn})

with dΦn/n+1 the n/n + 1 particle phase space, Θ(n/n+1) obs. measure function

formal requirements on the measurement function:Θ(n+1)(p1, . . . , pi = λq, . . . , pn+1) → Θ(n)(p1, . . . , pn+1)

for λ→ 0 soft limitΘ(n+1)(p1, . . . , pi , . . . , pj , . . . , pn+1) → Θ(n)(p1, . . . , p, . . . , pn+1)

for pi → zp , pj = (1− z)p collinear limitsimply remove (soft) parton i , or replace collinear pair {pi , pj } by p = pi + pj

; definition of infrared & collinear safe observables

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Hard Processes at Next-to-Leading Order QCDInfrared & Collinear safe observables

interested in observable-independent formulation

OB =∫|M(n)


OR =∫|M(n+1)

R |2 dΦn+1Θ(n+1)(O, {pn+1})

OV =∫

2<(MBM†V) dΦnΘ(n)(O, {pn})

with dΦn/n+1 the n/n + 1 particle phase space, Θ(n/n+1) obs. measure functionformal requirements on the measurement function:

Θ(n+1)(p1, . . . , pi = λq, . . . , pn+1) → Θ(n)(p1, . . . , pn+1)for λ→ 0 soft limit

Θ(n+1)(p1, . . . , pi , . . . , pj , . . . , pn+1) → Θ(n)(p1, . . . , p, . . . , pn+1)for pi → zp , pj = (1− z)p collinear limit

simply remove (soft) parton i , or replace collinear pair {pi , pj } by p = pi + pj

; definition of infrared & collinear safe observables5/30

Hard Processes at Next-to-Leading Order QCDAnatomy of NLO QCD calculations cont’d

σNLO2→n =

∫n

d(4)σB +∫

n+1d(d)σR +

∫n


finite corrections for IR safe observablesanalytical integration with cuts in d-dim infeasible for high-multi final stateswe are interested in fully-differential answer, use of MC integration methodshowever, both terms σR & σV exhibit divergences, i.e. are unboundσR in (n + 1)-particle phase space, σV in n-particle phase space

Subtraction methodsattempt to devise local subtraction term such that both integrals areseparately finite and can be performed as ordinary multi-dimensionalphase-space integrals; based on universality of QCD IR divergences; aim for IR regularisation at integrand level

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Hard Processes at Next-to-Leading Order QCDAnatomy of NLO QCD calculations cont’d

σNLO2→n =

∫n

d(4)σB +∫

n+1d(d)σR +

∫n


finite corrections for IR safe observablesanalytical integration with cuts in d-dim infeasible for high-multi final stateswe are interested in fully-differential answer, use of MC integration methodshowever, both terms σR & σV exhibit divergences, i.e. are unboundσR in (n + 1)-particle phase space, σV in n-particle phase space

Subtraction methodsattempt to devise local subtraction term such that both integrals areseparately finite and can be performed as ordinary multi-dimensionalphase-space integrals; based on universality of QCD IR divergences; aim for IR regularisation at integrand level

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Hard Processes at Next-to-Leading Order QCDSubtraction methods cont’d

subtraction terms provide local approximation for the real-emission process∫n+1

d(d)σA =∫

n

∫1

d(d)σA

↪→ captures soft & collinear limits of amplitudes [1/ε and 1/ε2 poles]

↪→ analytically integrable (d-dim) over one-parton emission phase space

add & subtract from NLO differential cross section

σNLO2→n =

∫n+1

[d(4)σR−d(4)σA

]+∫

n

[d(4)σB +

∫loop

d(d)σV+∫

1d(d)σA

]ε=0

; separately finite phase-space integrals (4-dim); can be performed by means of MC integration; no ambiguous or unphysical cut-offs/scales get introduced

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d(d)σA =∫

n

∫1

d(d)σA




σNLO2→n =

∫n+1

[d(4)σR−d(4)σA

]+∫

n

[d(4)σB +

∫loop

d(d)σV+∫

1d(d)σA

]ε=0


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d(d)σA =∫

n

∫1

d(d)σA




σNLO2→n =

∫n+1

[d(4)σR−d(4)σA

]+∫

n

[d(4)σB +

∫loop

d(d)σV+∫

1d(d)σA

]ε=0


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Hard Processes at Next-to-Leading Order QCDDipole subtraction method [Catani, Seymour Nucl. Phys. B 485 (1997) 291]

Catani & Seymour presented general expression for d(d)σA

(known as the dipole factorisation formula)constructed from Born process using universal dipole terms∫

n+1d(d)σA =

∑dipoles

∫n

d(d)σB ⊗∫

1d(d)Vdipole =

∫m

[d(d)σB ⊗ I

]↪→ universal dipole terms←↩spin- & color correlations

sum over dipoles contains all real-emission soft/collinear divergencessuited for any process with massless partons (0, 1, 2 initial-state hadrons)generalisation to massive partons [Catani,Dittmaier, Seymour, Trocsanyi Nucl. Phys. B 627 (2002) 189]

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Hard Processes at Next-to-Leading Order QCDDipole subtraction method cont’d

subtraction method for arbitrary observables (IRC safe)

OB =∫|M(n)


ORS =∫

dΦn+1

[|M(n+1)

R ({pn})|2 Θ(n+1)(O, {pn+1})

−∑

i 6=j 6=kDij,k({pn}) Θ(n)(O, p1, .., pij , pk , .., pn+1})

]OVI =

∫dΦn

[2<(MBM†V) +

∫1[dp]Dij,k

]Θ(n)(O, {pn})

[dp] phase-space element of one-partonsuch, differential & integrated subtraction terms cancel exactly↪→ requires mapping of (n + 1)→ n configuration↪→ need to distinguish different dipole types

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Hard Processes at Next-to-Leading Order QCDDipole subtraction method cont’d

dipoles involve three partons, splitting particles i , j and spectator kdistinguish initial- & final-state splitter/spectatorall contributions known, suitable for automated evaluation

FF FI

IF II

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Example: e+e− → qq @ NLO QCDmost simple scattering processsensitive to QCD colour charges & strong coupling αS

first non-trivial contribution to jet sub-structure; stringent test of QCD dynamics

Born-level contribution: (e+e− →)γ∗ → qq

Mqq =

p1

p2

ie γµ

= ua(p1)ieqγµδabvb(p2)

σBqq = σLO

qq ∝ |Mqq|2 = 4πα2

3Q2 e2qNc

where Q2 = (p1 + p2)2

q

q

11/30

Example: e+e− → qq @ NLO QCDmost simple scattering processsensitive to QCD colour charges & strong coupling αS

first non-trivial contribution to jet sub-structure; stringent test of QCD dynamics

Born-level contribution: (e+e− →)γ∗ → qq

Mqq =

p1

p2

ie γµ

= ua(p1)ieqγµδabvb(p2)

σBqq = σLO

qq ∝ |Mqq|2 = 4πα2

3Q2 e2qNc

where Q2 = (p1 + p2)2

q

q

11/30

Example: e+e− → qq @ NLO QCDreal-emission correction, i.e. (e+e− →)γ∗ → qqg

Mqqg = k ,ε ie γ

µ

p1

p2

+k ,ε

ie γµ

p1

p2

= −ua(p1)igs ε/tAab

i(/p1 + /k)(p1 + k)2 ieqγµvb(p2) + ua(p1)ieqγµ

i(/p2 + /k)(p2 + k)2 igs ε/tA

abvb(p2)

defining Q2 = (p1 + p2 + k)2 = s and xi = 2pi · Q/Q2, we find

|Mqqg |2 = CF8παSQ2

x21 + x2

2(1− x1)(1− x2) |Mqq|2 ; singular when x1/2 → 1

associated phase-space element

dΦ(3) = Q2

16π2 dx1dx2 Θ(1− x1)Θ(1− x2)Θ(x1 + x2 − 1)

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Example: e+e− → qq @ NLO QCDreal-emission correction, i.e. (e+e− →)γ∗ → qqg

Mqqg = k ,ε ie γ

µ

p1

p2

+k ,ε

ie γµ

p1

p2

= −ua(p1)igs ε/tAab

i(/p1 + /k)(p1 + k)2 ieqγµvb(p2) + ua(p1)ieqγµ

i(/p2 + /k)(p2 + k)2 igs ε/tA

abvb(p2)

defining Q2 = (p1 + p2 + k)2 = s and xi = 2pi · Q/Q2, we find

|Mqqg |2 = CF8παSQ2

x21 + x2

2(1− x1)(1− x2) |Mqq|2 ; singular when x1/2 → 1

associated phase-space element

dΦ(3) = Q2

16π2 dx1dx2 Θ(1− x1)Θ(1− x2)Θ(x1 + x2 − 1)

12/30

Example: e+e− → qq @ NLO QCDreal-emission subtraction term there are two FF dipoles contributing

D13,2(p1, p2, k) = 12p1k Vqg,q |Mqq|2

D23,1(p1, p2, k) = 12p2k Vqg,q |Mqq|2

corresponding subtracted real-emission cross section (d = 4)

σRSqqg =

∫3

[dσR

ε=0 − dσAε=0] (

pij + pk = Q , pk = 1xk

pk , pij = Q − 1xk

pk

)= |Mqq|2

CFαs

2π

∫ 1

0dx1 dx2 Θ(x1 + x2 − 1)

{[

x21 + x2

2(1− x1)(1− x2)

]Θ(3)(p1, p2, p3)

−[ 1

1− x2

( 22− x1 − x2

− (1 + x1))

+ 1− x1

x2

]Θ(2)(p13, p2)

−[ 1

1− x1

( 22− x1 − x2

− (1 + x2))

+ 1− x2

x1

]Θ(2)(p23, p1)

}; finite as x1/2 → 1 (implying Θ(3) → Θ(2))

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Example: e+e− → qq @ NLO QCDvirtual correction & integrated dipole contribution (d = 4− 2ε)

V ⊃−ie γ

µie γ

µ

x

|M(d)qq |

2(1−loop)

MS= |M(d)qq |

2 CFαs

2π1

Γ(1− ε)

(4πµ2

Q2

)ε {− 2ε2 −

3ε− 8 + π2 +O(ε)

}

integrated dipole kernels

I(ε) = CFαs

2π1

Γ(1− ε)

(4πµ2

Q2

)ε { 2ε2 + 3

ε+ 10− π2 +O(ε)

}combining both contributions yields

σVIqq =

∫2

[dσV

qq +∫

1dσA

qq

]ε=0

= |Mqq|2CFαs

π

∫dy12 δ(1− y12) Θ(2)(p1, p2)

where y12 = 2p1 · p2/Q2

14/30


V ⊃−ie γ

µie γ

µ

x

|M(d)qq |

2(1−loop)

MS= |M(d)qq |

2 CFαs

2π1

Γ(1− ε)

(4πµ2

Q2

)ε {− 2ε2 −

3ε− 8 + π2 +O(ε)

}integrated dipole kernels

I(ε) = CFαs

2π1

Γ(1− ε)

(4πµ2

Q2

)ε { 2ε2 + 3

ε+ 10− π2 +O(ε)

}

combining both contributions yields

σVIqq =

∫2

[dσV

qq +∫

1dσA

qq

]ε=0

= |Mqq|2CFαs

π

∫dy12 δ(1− y12) Θ(2)(p1, p2)

where y12 = 2p1 · p2/Q2

14/30


V ⊃−ie γ

µie γ

µ

x

|M(d)qq |

2(1−loop)

MS= |M(d)qq |

2 CFαs

2π1

Γ(1− ε)

(4πµ2

Q2

)ε {− 2ε2 −

3ε− 8 + π2 +O(ε)

}integrated dipole kernels

I(ε) = CFαs

2π1

Γ(1− ε)

(4πµ2

Q2

)ε { 2ε2 + 3

ε+ 10− π2 +O(ε)

}combining both contributions yields

σVIqq =

∫2

[dσV

qq +∫

1dσA

qq

]ε=0

= |Mqq|2CFαs

π

∫dy12 δ(1− y12) Θ(2)(p1, p2)

where y12 = 2p1 · p2/Q2

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Example: e+e− → qq @ NLO QCD

total scattering cross sectionmeasurement functions given by Θ(3) = Θ(2) = 1

σNLOqq =

∫3

dσRSqqg +

∫2

[dσB

qq + dσVIqq]

= σLOqq

(1 + 3

4CFαs

π

)+O(α2

s )

Discussion:fully inclusive total cross section [IR safe]; real & virtual IR singularities properly cancelledyields modest, finite correctiondependent on strong-coupling parameter and its running

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Example: e+e− → qq @ NLO QCDThe emerging picture

corrections to σtot dominated by hard, large-angle gluonssoft gluons play no role for σtot

collision characterised by thard ∼ 1/Qsoft-gluons emitted on long time scales tsoft ∼ 1/(Eθ2); cannot influence cross sectiontransition to hadrons occurs on long time scales thad ∼ 1/ΛQCD; can thus be ignored

with proper choice for scale of αs , µ2 = Q2, perturbation theory works well

σtot = σqq

1︸︷︷︸LO

+1.045αs(Q2)π︸︷︷︸

NLO

+0.94(αs(Q2)π

)2

︸︷︷︸NNLO

−15(αs(Q2)π

)3

︸︷︷︸NNNLO

+ · · ·

[coefficients given for Q2 = M2

Z , including EW corrections]

Total cross sections are inclusive quantities,inclusive in the number of additional QCD partons!

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Scale Uncertainties of Fixed-Order PredictionsThe scale ambiguity

fixed-order prediction depends on (arbitrary) renormalisation scale µestimate theory uncertainty through scale variations, e.g. Q

2 < µ < 2Q; provides estimate of uncalculated higher-order terms

σNLOtot (µ2) = σqq

(1 + C1αs(µ2)

)withαs(µ2) = αs(Q2)− 2b0α

2s (Q2) ln

(µ

Q

)+O(α3

s )

= σqq

(1 + C1αs(Q2)− 2C1b0α

2s (Q2) ln

(µ

Q

)+O(α3

s ))

compare to

σNNLOtot (µ2) = σqq

(1 + C1αs(µ2) + C2(µ2)α2

s (µ2))

with C2(µ2) = C2(Q2) + 2C1b0α2s (Q2) ln

(µ

Q

)

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Scale Uncertainties of Fixed-Order Predictions

The scale ambiguity

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

0.1 1 10

σee →

hadro

ns /

σee →

qq

µR / Q

scale-dep. of σ(e+e

- → hadrons)

Q = MZ

0.5 < xµ < 2

conventional range

LO

NLO

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

0.1 1 10σ

ee →

hadro

ns /

σee →

qq

µR / Q

scale-dep. of σ(e+e

- → hadrons)

Q = MZ

0.5 < xµ < 2

conventional range

LO

NLO

NNLO

; use scale variations to estimate theory uncertainty; probe residual dependence on (uncalculated) higher orders

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NLO QCD: brief summaryMotivations

accurate cross-section estimatesreal-emission kinematics correctionsreduced systematic uncertainties [scale dependences]

Anatomy

↪→ subtraction of infrared singularities in real- & virtual corrections[Catani, Seymour Nucl. Phys. B 485 (1997) 291, Frixione et al. Nucl. Phys. B 467 (1996) 399]

↪→ dedicated one-loop amplitude codes: BLACKHAT, OPENLOOPS/COLLIER, RECOLA, ...

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NLO QCD: brief summaryMotivations

accurate cross-section estimatesreal-emission kinematics correctionsreduced systematic uncertainties [scale dependences]

Anatomy

↪→ subtraction of infrared singularities in real- & virtual corrections[Catani, Seymour Nucl. Phys. B 485 (1997) 291, Frixione et al. Nucl. Phys. B 467 (1996) 399]

↪→ dedicated one-loop amplitude codes: BLACKHAT, OPENLOOPS/COLLIER, RECOLA, ...

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Interlude: QCD jets & PDFs

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The emergent picture: final-state jets

Jet definition (prel.): jets are collimated sprays of hadronic particles

hard partons undergo soft and collinear showeringhadrons closely correlated with the hard partons’ directions

q

q

Counting jets; near perfect two-jet event; almost all energy contained in two cones

21/30

The emergent picture: final-state jets

Jet definition (prel.): jets are collimated sprays of particles

hard partons undergo soft and collinear showeringhadrons closely correlated with the hard partons’ directions

Counting jets; hard emissions can induce more jets; jet counting not obvious, is this a three- or four-jet event?

22/30

Defining jetsJet definition (addendum): jet number shouldn’t depend upon just a

soft/collinear emission

; Infrared & collinear safety

jet 1 jet 2

LO partons

Jet Def n

jet 1 jet 2

Jet Def n

NLO partons

jet 1 jet 2

Jet Def n

parton shower

jet 1 jet 2

Jet Def n

hadron level

π π

K

p φ

Infrared & Collinear safe jet definitionscrucial for comparing theory with experimental results

23/30

Jet algorithms

Jet definitiongroup together particles into a common jets [jet algorithm]

typical parameter is R, distance in y − φ space, determines angular reachcombine momenta of jet constituents to yield jet momentum [recombination scheme]

two generic types of jet algorithms are commonly used:cone algorithms

widely used in the past at the Tevatronjets have regular/circular shapessome suffer from IR or collinear unsafety

sequential recombination algorithmswidely used at LEP [Durham kT algorithm]

jet can have irregular shapesdefault at the LHC experiments [anti-kT algorithm]

24/30

Sequential recombination algorithms

A generic jet finding algorithm1 compute a distance measure yij for each pair of final-state particles2 determine all distance measures wrt the beam yiB3 determine the minimum of all yij ’s and yiB ’s

1 if yij is smallest, combine particles ij, sum four-momenta2 if yiB is smallest, remove particle i , call it a jet

4 go back to step one, until all particles are clustered into jetsin analyses one typically uses

jets with inter-jet distances yij > ycut [exclusive mode]jets with inter-jet distances yij > ycut & E > Ecut [inclusive mode]

different algorithms use different measures: yij & yiB

25/30

Sequential recombination algorithms: the kT algorithm

recall the soft and collinear limit of the gluon-emission probability for a→ ij

dS '2αsCA/F

π

dEimin(Ei ,Ej)

dθijθij

,

using min(Ei ,Ej) we can avoid specifying which of i and j is soft

The kT -algorithm distance measure

yij =2 min(E 2

i ,E 2j )(1− cos θij)Q2

; in the collinear limit: yij ' min(E 2i E 2

j )θ2ij/Q2

; relative transverse momentum, normalized to total energy; soft/collinear particles get clustered first

26/30

Sequential recombination algorithms: the anti-kT algorithm

recall the soft and collinear limit of the gluon-emission probability for a→ ij

dS ' 2αsCiπ

dEimin(Ei ,Ej)

dθijθij

,

using min(Ei ,Ej) we can avoid specifying which of i and j is soft

The anti-kT -algorithm distance measure

yij = 2Q2 min(E−2i ,E−2

j )(1− cos θij)

; jet-finding starts out with hard objects; softer particles get clustered into hard jets later on; produces nicely regular shaped jets; default in current LHC physics analyses

27/30

Jet algorithms at work: kT jets at LEP

kT jet fractions at LEP

[Gehrmann-De Ridder et al. Phys. Rev. Lett. 100 (2008) 172001]

28/30

Jet algorithms at work: kT jets at LEP

kT jet fractions at LEP

[Gehrmann-De Ridder et al. Phys. Rev. Lett. 100 (2008) 172001]

28/30

Jet algorithms at work: anti-kT jets at LHC

anti-kT inclusive jets at LHC

29/30

Jet algorithms at work: anti-kT jets at LHCThe V+jets processesbackground to many BSM searcheswide range of kinematics↪→ multi-scale QCD problem

Njets: jet multiplicity[ATLAS-CONF-2016-046]

BH+SH: NLO QCD MEs

SHERPA: 0,1,2j MEPS@NLO + 3,4j MEPS@LO

FxFx: 0,1,2j NLO MEs + shower

30/30

Jet algorithms at work: anti-kT jets at LHCThe V+jets processesbackground to many BSM searcheswide range of kinematics↪→ multi-scale QCD problem

HT : scalar sum of jet pT ’s[ATLAS-CONF-2016-046]

30/30

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Introduction to Quantum Chromodynamics